Foundations

14 pages, 317 KiB

Open AccessArticle

Extended Convergence of Two Multi-Step Iterative Methods

by Samundra Regmi, Ioannis K. Argyros, Jinny Ann John and Jayakumar Jayaraman

Foundations 2023, 3(1), 140-153; https://doi.org/10.3390/foundations3010013 - 13 Mar 2023

Cited by 2 | Viewed by 1251

Iterative methods which have high convergence order are crucial in computational mathematics since the iterates produce sequences converging to the root of a non-linear equation. A plethora of applications in chemistry and physics require the solution of non-linear equations in abstract spaces iteratively. [...] Read more.

Iterative methods which have high convergence order are crucial in computational mathematics since the iterates produce sequences converging to the root of a non-linear equation. A plethora of applications in chemistry and physics require the solution of non-linear equations in abstract spaces iteratively. The derivation of the order of the iterative methods requires expansions using Taylor series formula and higher-order derivatives not present in the method. Thus, these results cannot prove the convergence of the iterative method in these cases when such higher-order derivatives are non-existent. However, these methods may still converge. Our motivation originates from the need to handle these problems. No error estimates are given that are controlled by constants. The process introduced in this paper discusses both the local and the semi-local convergence analysis of two step fifth and multi-step

5 + 3 r

order iterative methods obtained using only information from the operators on these methods. Finally, the novelty of our process relates to the fact that the convergence conditions depend only on the functions and operators which are present in the methods. Thus, the applicability is extended to these methods. Numerical applications complement the theory. Full article

(This article belongs to the Collection Editorial Board Members’ Collection Series: Theory and Its Applications in Problems of Mathematical Physics and of Mathematical Chemistry)

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13 pages, 308 KiB

Open AccessArticle

Extended Convergence for Two Sixth Order Methods under the Same Weak Conditions

by Ioannis K. Argyros, Samundra Regmi, Jinny Ann John and Jayakumar Jayaraman

Foundations 2023, 3(1), 127-139; https://doi.org/10.3390/foundations3010012 - 10 Mar 2023

Cited by 3 | Viewed by 1046

Abstract

High-convergence order iterative methods play a major role in scientific, computational and engineering mathematics, as they produce sequences that converge and thereby provide solutions to nonlinear equations. The convergence order is calculated using Taylor Series extensions, which require the existence and computation of [...] Read more.

High-convergence order iterative methods play a major role in scientific, computational and engineering mathematics, as they produce sequences that converge and thereby provide solutions to nonlinear equations. The convergence order is calculated using Taylor Series extensions, which require the existence and computation of high-order derivatives that do not occur in the methodology. These results cannot, therefore, ensure that the method converges in cases where there are no such high-order derivatives. However, the method could converge. In this paper, we are developing a process in which both the local and semi-local convergence analyses of two related methods of the sixth order are obtained exclusively from information provided by the operators in the method. Numeric applications supplement the theory. Full article

(This article belongs to the Section Mathematical Sciences)

12 pages, 4452 KiB

Open AccessArticle

Lagrangians of Multiannual Growth Systems

by Petri P. Kärenlampi

Foundations 2023, 3(1), 115-126; https://doi.org/10.3390/foundations3010011 - 9 Mar 2023

Viewed by 1352

Abstract

Multiannual growth systems are modeled in generic terms and investigated using partial derivatives and Lagrange multipliers. Grown stock density and temperature sum are used as independent variables. Estate capitalization increases continuously with grown stock and temperature sum, whereas capital return rate and gross [...] Read more.

Multiannual growth systems are modeled in generic terms and investigated using partial derivatives and Lagrange multipliers. Grown stock density and temperature sum are used as independent variables. Estate capitalization increases continuously with grown stock and temperature sum, whereas capital return rate and gross profit rate reach a maximum with respect to grown stock. As two restrictions are applied simultaneously, the results mostly but not always follow intuition. The derivative of capital return rate with respect to gross profit rate is positive, and negative with respect to capitalization. The derivative of capitalization with respect to capital return rate shows some positive values, as well as that with respect to gross profit rate. The derivative of the gross profit rate is positive with respect to both capitalization and capital return rate. The results indicate a variety of alternative strategies, which may or may not be multiobjective. Full article

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16 pages, 863 KiB

Open AccessArticle

Fatty Acid Vesicles as Hard UV-C Shields for Early Life

by Iván Lechuga and Karo Michaelian

Foundations 2023, 3(1), 99-114; https://doi.org/10.3390/foundations3010010 - 23 Feb 2023

Cited by 2 | Viewed by 2287

Abstract

Theories on life’s origin generally acknowledge the advantage of a semi-permeable vesicle (protocell) for enhancing the chemical reaction–diffusion processes involved in abiogenesis. However, more and more evidence indicates that the origin of life is concerned with the photo-chemical dissipative structuring of the fundamental [...] Read more.

Theories on life’s origin generally acknowledge the advantage of a semi-permeable vesicle (protocell) for enhancing the chemical reaction–diffusion processes involved in abiogenesis. However, more and more evidence indicates that the origin of life is concerned with the photo-chemical dissipative structuring of the fundamental molecules under soft UV-C light (245–275 nm). In this paper, we analyze the Mie UV scattering properties of such a vesicle created with long-chain fatty acids. We find that the vesicle could have provided early life with a shield from the faint but destructive hard UV-C ionizing light (180–210 nm) that probably bathed Earth’s surface from before the origin of life and at least until 1200 million years after, until the formation of a protective ozone layer as a result of the evolution of oxygenic photosynthesis. Full article

(This article belongs to the Special Issue The Non-equilibrium Thermodynamic Foundations of Fundamental Problems from Nature)

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17 pages, 823 KiB

Open AccessArticle

Extended Newton-like Midpoint Method for Solving Equations in Banach Space

by Ioannis K. Argyros, Gagan Deep and Samundra Regmi

Foundations 2023, 3(1), 82-98; https://doi.org/10.3390/foundations3010009 - 21 Feb 2023

Cited by 3 | Viewed by 1698

Abstract

In this study, we present a convergence analysis of a Newton-like midpoint method for solving nonlinear equations in a Banach space setting. The semilocal convergence is analyzed in two different ways. The first one is shown by replacing the existing conditions with weaker [...] Read more.

In this study, we present a convergence analysis of a Newton-like midpoint method for solving nonlinear equations in a Banach space setting. The semilocal convergence is analyzed in two different ways. The first one is shown by replacing the existing conditions with weaker and tighter continuity conditions, thereby enhancing its applicability. The second one uses more general

ω

-continuity conditions and the majorizing principle. This approach includes only the first order Fréchet derivative and is applicable for problems that were otherwise hard to solve by using approaches seen in the literature. Moreover, the local convergence is established along with the existence and uniqueness region of the solution. The method is useful for solving Engineering and Applied Science problems. The paper ends with numerical examples that show the applicability of our convergence theorems in cases not covered in earlier studies. Full article

(This article belongs to the Section Mathematical Sciences)

10 pages, 2828 KiB

Open AccessPerspective

Investigation of the Acid/Basic Sites of Zeolite Trough Some Catalysed Nucleophilic Reactions

by Valentina Verdoliva, Michele Saviano and Stefania De Luca

Foundations 2023, 3(1), 72-81; https://doi.org/10.3390/foundations3010008 - 16 Feb 2023

Viewed by 2087

Abstract

Zeolites, both natural and synthetic, are certainly some of the most versatile minerals for their applications. Since the 1940s, they have been used in the chemical industry as catalysts, adsorbents and ion exchanger extensively, and the development of their practical usage is expected [...] Read more.

Zeolites, both natural and synthetic, are certainly some of the most versatile minerals for their applications. Since the 1940s, they have been used in the chemical industry as catalysts, adsorbents and ion exchanger extensively, and the development of their practical usage is expected to continue upon years. Their versatility is the result of the combination of peculiar and indispensable properties, each of which can be found in other material as a single property, but seldom all of them are found in combination. However, despite the success of their employment, the mechanisms of many important catalytic processes involving zeolites remained elusive. In particular, the comprehension of the structure–property relationships for emerging applications are highly required. In this perspective article we focus on the role of zeolites as solid acid-base catalysts. We go deeply into the structural properties of the LTA kind (Zeolite-Na A 4 Ångstrom) that was successfully employed as basic catalyst for several nucleophilic substitution reactions. Full article

(This article belongs to the Special Issue Solid Catalysts for Chemical Processes)

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7 pages, 986 KiB

Open AccessCommunication

Zero-Energy Bound States of Neutron–Neutron or Neutron–Muon Systems

by Eugene Oks

Foundations 2023, 3(1), 65-71; https://doi.org/10.3390/foundations3010007 - 6 Feb 2023

Viewed by 1334

Abstract

There exists the following paradigm: for interaction potentials U(r) that are negative and go to zero as r goes to infinity, bound states may exist only for the negative total energy E. For E > 0 and for E = 0, [...] Read more.

There exists the following paradigm: for interaction potentials U(r) that are negative and go to zero as r goes to infinity, bound states may exist only for the negative total energy E. For E > 0 and for E = 0, bound states are considered to be impossible, both in classical and quantum mechanics. In the present paper we break this paradigm. Namely, we demonstrate the existence of bound states of E = 0 in neutron–neutron systems and in neutron–muon systems, specifically when the magnetic moments of the two particles in the pair are parallel to each other. As particular examples, we calculate the root-mean-square size of the bound states of these systems for the values of the lowest admissible values of the angular momentum, and show that it exceeds the neutron radius by an order of magnitude. We also estimate the average kinetic energy and demonstrate that it is nonrelativistic. The corresponding bound states of E = 0 may be called “neutronium” (for the neutron–neutron systems) and “neutron–muonic atoms” (for the neutron–muon systems). We also point out that this physical system possesses higher-than-geometric (i.e., algebraic) symmetry, leading to the approximate conservation of the square of the angular momentum, despite the geometric symmetry being axial. We use this fact for facilitating analytical and numerical calculations. Full article

(This article belongs to the Special Issue Advances in Fundamental Physics II)

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2 pages, 196 KiB

Open AccessEditorial

Acknowledgment to the Reviewers of Foundations in 2022

by Foundations Editorial Office

Foundations 2023, 3(1), 63-64; https://doi.org/10.3390/foundations3010006 - 20 Jan 2023

Viewed by 1045

Abstract

High-quality academic publishing is built on rigorous peer review [...] Full article

14 pages, 313 KiB

Open AccessArticle

New Generalized Hermite–Hadamard–Mercer’s Type Inequalities Using (k, ψ)-Proportional Fractional Integral Operator

by Henok Desalegn Desta, Eze R. Nwaeze, Tadesse Abdi and Jebessa B. Mijena

Foundations 2023, 3(1), 49-62; https://doi.org/10.3390/foundations3010005 - 11 Jan 2023

Cited by 2 | Viewed by 1308

Abstract

In this paper, by using Jensen–Mercer’s inequality we obtain Hermite–Hadamard–Mercer’s type inequalities for a convex function employing left-sided

(k, ψ)

-proportional fractional integral operators involving continuous strictly increasing function. Our findings are a generalization of some results that existed in the literature. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)

12 pages, 269 KiB

Open AccessArticle

Treatment of a Coupled System for Quadratic Functional Integral Equation on the Real Half-Line via Measure of Noncompactness

by Ahmed M. A. El-Sayed, Yasmin M. Y. Omar, Hind H. G. Hashem and Shorouk M. Al-Issa

Foundations 2023, 3(1), 37-48; https://doi.org/10.3390/foundations3010004 - 6 Jan 2023

Cited by 1 | Viewed by 1357

Abstract

This article is devoted to the solvability and the asymptotic stability of a coupled system of a functional integral equation on the real half-axis. Our consideration is located in the space of bounded continuous functions on

R_{+}

[...] Read more.

This article is devoted to the solvability and the asymptotic stability of a coupled system of a functional integral equation on the real half-axis. Our consideration is located in the space of bounded continuous functions on

R_{+}

(B C (R_{+})) .

The main tool applied in this work is the technique associated with measures of noncompactness in

B C (R_{+})

by a given modulus of continuity. Next, we formulate and prove a sufficient condition for the solvability of that coupled system. We, additionally, provide an example and some particular cases to demonstrate the effectiveness and value of our results. Full article

(This article belongs to the Section Mathematical Sciences)

12 pages, 781 KiB

Open AccessArticle

Improved Higher Order Compositions for Nonlinear Equations

by Gagan Deep and Ioannis K. Argyros

Foundations 2023, 3(1), 25-36; https://doi.org/10.3390/foundations3010003 - 6 Jan 2023

Cited by 1 | Viewed by 1708

Abstract

In the present study, two new compositions of convergence order six are presented for solving nonlinear equations. The first method is obtained from the third-order one given by Homeier using linear interpolation, and the second one is obtained from the third-order method given [...] Read more.

In the present study, two new compositions of convergence order six are presented for solving nonlinear equations. The first method is obtained from the third-order one given by Homeier using linear interpolation, and the second one is obtained from the third-order method given by Traub using divided differences. The first method requires three evaluations of the function and one evaluation of the first derivative, thereby enhancing the efficiency index. In the second method, the computation of a derivative is reduced by approximating it using divided differences. Various numerical experiments are performed which demonstrate the accuracy and efficacy of the proposed methods. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences II)

9 pages, 599 KiB

Open AccessBrief Report

Methylidyne Cavity Ring-Down Spectroscopy in a Microwave Plasma Discharge

by László Nemes and Christian G. Parigger

Foundations 2023, 3(1), 16-24; https://doi.org/10.3390/foundations3010002 - 5 Jan 2023

Viewed by 1702

Abstract

This work communicates cavity ring-down spectroscopy (CRDS) of methylidyne (CH) in a chemiluminescent plasma that is produced in a microwave cavity. Of interest are the rotational lines of the 0-0 vibrational transition for the A–X band and the 1-0 vibrational transition for the [...] Read more.

This work communicates cavity ring-down spectroscopy (CRDS) of methylidyne (CH) in a chemiluminescent plasma that is produced in a microwave cavity. Of interest are the rotational lines of the 0-0 vibrational transition for the A–X band and the 1-0 vibrational transition for the B–X band. The reported investigations originate from research on the CH radical in 1996, which constituted the first case of applying CRDS to the CH radical. The report also includes a recent analysis that shows excellent agreement of the measured and computed data, and it communicates CH line strength data. The CH radical is an important diatomic molecule in hydrocarbon combustion diagnosis and the analysis of stellar plasma emissions, to name just two examples of analytical plasma chemistry. Full article

(This article belongs to the Special Issue Advances in Fundamental Physics II)

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15 pages, 1381 KiB

Open AccessTechnical Note

Diatomic Line Strengths for Fitting Selected Molecular Transitions of AlO, C₂, CN, OH,

N_{2}^{+}

, NO, and TiO, Spectra

by Christian G. Parigger

Foundations 2023, 3(1), 1-15; https://doi.org/10.3390/foundations3010001 - 1 Jan 2023

Cited by 4 | Viewed by 2655

Abstract

This work communicates line-strength data and associated scripts for the computation and spectroscopic fitting of selected transitions of diatomic molecules. The scripts for data analysis are designed for inclusion in various software packages or program languages. Selected results demonstrate the applicability of the [...] Read more.

This work communicates line-strength data and associated scripts for the computation and spectroscopic fitting of selected transitions of diatomic molecules. The scripts for data analysis are designed for inclusion in various software packages or program languages. Selected results demonstrate the applicability of the program for data analysis in laser-induced optical breakdown spectroscopy primarily at the University of Tennessee Space Institute, Center for Laser Applications. Representative spectra are calculated and referenced to measured data records. Comparisons of experiment data with predictions from other tabulated diatomic molecular databases confirm the accuracy of the communicated line-strength data. Full article

(This article belongs to the Special Issue Advances in Fundamental Physics II)

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Foundations, Volume 3, Issue 1 (March 2023) – 13 articles

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